Conte's need is the greater. Having missed out on Champions League football for next season, many expect the Wembley showpiece to be the Italian's final match in charge at Stamford Bridge, regardless of the outcome.
If it is, nothing will give Conte more joy than bowing out by winning a cup competition for the first time as a coach by beating the man with whom he has clashed most regularly and bitterly since arriving in England.
Conte labelled Mourinho a "little man" and "fake" at the peak of their spat in December.
Days earlier, Mourinho had appeared to allude to a suspension given to Conte for allegedly failing to report match-fixing when coaching at Siena. Conte later won an appeal, clearing him of any wrongdoing in the case.
The root of their feud comes from Conte's success in his first campaign at Chelsea, just a season after Mourinho's second spell at Stamford Bridge ended in the Portuguese being sacked having won just four of Chelsea's first 16 league games, in December 2015.
On his first return as a visiting manager, Mourinho protested at what he perceived as Conte's "humiliating" celebrations in a 4-0 victory for the Blues over United.
Conte then warned his players at the start of this season not to let their standards slip to the levels of "the last season with Mourinho".
- Chelsea slip -
In hindsight, the Italian was right to warn against complacency as Chelsea have fallen well short of the form they showed in storming to the title last season.
A fifth-placed finish in the Premier League was confirmed when they capitulated 3-0 to Newcastle on the final day of the season on Sunday.
"To finish fifth is a big disappointment. How damaging is it not to be in the Champions League? You'll have to ask the club about that," Conte said, only intensifying speculation about his future.
"If we play like this in the FA Cup final, we don't have a chance. We have six days to change our approach, our desire, our will to fight because we can do much better."
Mourinho has at least ensured Champions League qualification for next season via a second-place finish in the Premier League.
However, a record 19-point gap to champions Manchester City and a deeply disappointing Champions League last-16 exit to Sevilla means the Portuguese is also left hanging on until the final game of the season to deliver a trophy that would silence some of the critics of United's often turgid performances, lacking in the invention and flair displayed by rivals City and Champions League finalists Liverpool.
"Because I almost do that all the time (win a trophy), when I don't do it, you (the media) kill me," said Mourinho on the prospect of going without a trophy in his second season at Old Trafford. "Which is what you are all ready to do."
To do so, Mourinho must shake United out of their own end-of-season slumber -- the Red Devils have scored just once in their past three games.
Whether Mourinho can close the gap on City next season will depend on his ability to finally get the best out of his attacking talent, particularly marquee signings Paul Pogba and Alexis Sanchez.
Starting that process with a trophy on Saturday would be a strong statement of intent from United, and give Mourinho the extra satisfaction of showing Conte how short-lived life as Chelsea manager under ruthless Russian owner Roman Abramovich can be.